8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
1/99
INTERNAL ROTATION
IN SYMMETRIC TOP
MOLECULES
JYRKISCHRODERUS
Faculty of Science,
Department of Physical Sciences,University of Oulu
OULU 2004
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
2/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
3/99
JYRKI SCHRODERUS
INTERNAL ROTATION
IN SYMMETRIC TOP
MOLECULES
Academic Dissertation to be presented with the assent of
the Faculty of Science, University of Oulu, for public
discussion in Auditorium GO101, Linnanmaa, on
November 19th, 2004, at 12 noon
OULUN YLIOPISTO, OULU 2004
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
4/99
Copyright 2004
University of Oulu, 2004
Supervised byProfessor Seppo AlankoProfessor Rauno Anttila
Reviewed byProfessor Lauri HalonenProfessor Carlo di Lauro
ISBN 951-42-7566-7 (nid.)ISBN 951-42-7567-5 (PDF) http://herkules.oulu.fi/isbn9514275675/
ISSN 0355-3191 http://herkules.oulu.fi/issn03553191/
OULU UNIVERSITY PRESS
OULU 2004
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
5/99
Schroderus, Jyrki, Internal rotation in symmetric top molecules.
Faculty of Science, Department of Physical Sciences, University of Oulu, P.O.Box 3000, FIN-
90014 University of Oulu, Finland
2004
Oulu, Finland
Abstract
Internal rotation in symmetric top molecules offers an excellent opportunity to investigate large
amplitude motion in a relatively simple intramolecular environment. Due to specific symmetry
characteristics of a symmetric top molecular frame, the internal rotation degree of freedom is in the
zeroth order approximation separable from the small amplitude vibrations and the overall rotation,
thus enabling to characterize the vibrational-torsional-rotational energy structure with a relatively
simple Hamiltonian. Lessons from symmetric internal rotor studies may be applied to more complex
systems, such as asymmetric internal rotors and macromolecules.
This thesis deals with internal rotation in CH3SiH3, CH3SiD3, CH3CF3 which have become a
prototype of symmetric internal rotors. The thesis presents high resolution vibration-torsion-rotation
spectra and detailed analysis of these molecules. Particular attention is focused on torsion-mediated
interactions, such as Coriolis-type interactions and Fermi-type interactions, coupling the internal
rotation and the small amplitude vibrational motion.
The studies show that the expansion of the data to the small amplitude vibrations and inclusion ofthe torsion-mediated interactions play a crucial role in order to obtain an appropriate characterization
of the vibrational-torsional-rotational energy level structure and physically meaningful molecular
parameters.
Keywords: high resolution, infrared spectroscopy, internal rotation, symmetric top
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
6/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
7/99
To My Family
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
8/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
9/99
Acknowledgments
Most of the work associated with this thesis has been carried out in the University of British
Columbia, Canada, where I was supervised by professor Irving Ozier whom I would like to
thank for placing the facilities at my disposal, giving me inspiring scientific challenges, and
creating an excellent atmosphere in the spectroscopy group in Vancouver. I also want to
thank professor Nasser Moazzen-Ahmadi for supervising me during my stay in Canada and
for the hospitality during my stay in the University of Lethbridge. I also had the priviledge
to work with Dr. Shixin Wang both in Vancouver and Oulu.
I am greatly indebted to my supervisor professor Rauno Anttila for his financial and
personal support. I am also grateful to professor Tapio Rantala for providing deep physical
insight into molecular problems.
I am also grateful to professor Seppo Alanko for reading the manuscript of this the-
sis and giving me an example of dedication and genuine enthusiasm to scientific work.
Furthermore, I wish to express my gratitude to docent Veli-Matti Horneman for his uncom-
promising and patient attitude in complex spectroscopic measurements.
Special thanks to docent Juha Vaara for valuable discussion on various topics in physics.
I would like to thank Mr. Markku Pkkil for revising the language of the manuscript.
This thesis is dedicated to my wife Anne and my children Jaakko, Vappu and Hilla who
constantly bring light into my life.
Oulu, October 2004 Jyrki Schroderus
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
10/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
11/99
List of Original Papers
The present thesis is based on the following papers which are referred to in the text by theirRoman numerals:
I Schroderus J, Moazzen-Ahmadi N & Ozier I (2000) The 12 Band of CH3SiD3. J MolSpectrosc 201:292-296.
II Ozier I, Schroderus J, Wang S-X, McRae GA, Gerry MCL, Vogelsanger B & Bauder
A (1998) The Forbidden Rotational Q-branch of CH3CF3: Torsional Properties and
(A1
A2) Splittings. J Mol Spectrosc 190:324-340.
III Schroderus J, Moazzen-Ahmadi N & Ozier I (2001) The Triad in the Region of the
Lowest-Frequency Parallel Fundamental Band (v5 = 1 0) of CH3SiH3: Fermi-typeInteractions and Giant Torsional Splittings. J Chem Phys 115:1392-1404.
IV Schroderus J, Horneman V-M, Johnson M S, Moazzen-Ahmadi N & Ozier I (2002)
High-Resolution Far-Infrared Torsional Spectrum of CH3SiD3 Using a Synchrotron
Source. J Mol Spectrosc 215:134-143.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
12/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
13/99
Contents
Abstract
Acknowledgments
List of Original Papers
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Overview and Background of the Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1. Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2. Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3. Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4. Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5. Remarks on the Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1. Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2. Coordinate system for a Symmetric Top with Two Three-Fold Coaxial
Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3. Classical Kinetic Energy of a Rotating and Vibrating Molecule . . . . . . . . . . . . . . 26
3.4. Watson Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1. Expansion of the Watson Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.2. Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.3. Rigid Rotor-Symmetric Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.4. Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5. Internal Rotor in Hindering Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5.1. Principal Axis Method (PAM): = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5.2. Internal Axis Method (IAM): = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3. Remarks on the Coordinate Conventions. The Hybrid Method . . . . . . . . . 45
3.5.4. Schrdinger Equation of a Hindered Internal Rotor . . . . . . . . . . . . . . . . . . 46
3.6. Molecular Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6.1. G18 Permutation Inversion Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.2. Symmetry of the Vibrational-Torsional-Rotational States . . . . . . . . . . . . . 55
3.7. Schrdinger Equation of a Vibrating and Rotating Molecule . . . . . . . . . . . . . . . . 62
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
14/99
3.7.1. Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7.2. Time-Independent Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7.3. Contact Transformation of the Vibration-Torsion-Rotation Operator . . . . . 68
3.7.4. Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.8. Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8.1. Two-step Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8.2. One-step Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8.3. Remarks on the Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 79
4. Spectrum Analyses and Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1. Identification of Spectral Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2. Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3. Paper I: The Lowest Fundamental Band of CH3SiD3 . . . . . . . . . . . . . . . . . . . . . . 83
4.3.1. Analysis of the GS/12 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.2. Results and Discussion of the GS/12 System of CH3SiD3 . . . . . . . . . . . . 85
4.4. Paper II: Ground State Rotational Analysis of CH3CF3 . . . . . . . . . . . . . . . . . . . . 86
4.4.1. Analysis of the Sextic Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.2. Results and Discussion of the Rotational Analysis of the GS in
CH3CF3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5. Paper III: Three-Band System of CH3SiH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.1. Data Set of CH3SiH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.2. Analysis of the GS/12/5 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5.3. Results and Discussion of the GS/12/5 System . . . . . . . . . . . . . . . . . . . 90
4.6. Paper IV: Torsional Analysis in CH3SiD3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6.1. Data Set of CH3SiD3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6.2. Analysis of the Torsional Bands in CH3SiD3 . . . . . . . . . . . . . . . . . . . . . . . 914.6.3. Results and Discussion of the Torsional Analysis of CH3SiD3 . . . . . . . . . 92
5. Discussion and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Reference
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
15/99
1 Introduction
This thesis deals with internal rotation in polar symmetric top molecules, such as methylsilanes CH3SiH3 and CH3SiD3, and trifluoroethane CH3CF3. Symmetric top molecules
with a single internal rotation degree of freedom about the molecular symmetry axis pro-
vide an excellent opportunity for studying large amplitude motion in a relatively simple
molecular environment. Due to specific symmetry characteristics of the symmetric top
molecular frame, the torsional degree of freedom is in the zeroth order approximation sep-
arable from the small amplitude vibrational degrees of freedom and the overall rotational
degrees of freedom, thus providing a simplification in the overall Hamiltonian and the en-
ergy level structure compared to those of more complex molecules, such as asymmetric
tops with internal rotation. Thereby, the symmetric top studies may provide important
knowledge for studies of large amplitude motion in asymmetric tops.
In molecular rotation-vibration spectroscopy, a gaseous molecular sample is subjected
to an oscillating electromagnetic field that interacts with molecular charge distribution and
gives rise to transitions from a molecular quantum state to another. In a transition, the
molecule either absorbs or emits a part of the radiation spectrum which corresponds to an
energy difference between molecular quantum states involved in transitions. The spectrum
is recorded by spectroscopic means and analysed by using a molecular model in order to
obtain information on molecular properties.
The internal rotation complicates the quantum mechanical treatment and the vibration-
rotation energy structure in many respects when compared with semi-rigid symmetric tops.
First, the torsional energy typically falls in between rotational and vibrational energy scale,
thus generating a dense vibrational-torsional overtone energy level structure. In addition,each vibration-rotation-torsion energy level is split into three torsional sub-levels, labeled
by = 1, 0, +1, thus increasing the complexity of the energy level system.The internal rotation degree of freedom gives rise to a great number of redundant Hamil-
tonian terms that are difficult to separate into the individual contributions according to their
physical origin. The redundant terms primarily arise from two mechanisms.
First, to first order, the electric dipole selection rules (vT = k ` = = 0) ofthe vibrational-torsional-rotational transitions forbid direct observations between different
(vT, k, ) states, and therefore, the axial rotational parameters such as A0, D0,k, and theleading torsional parameters F0 and V0,3 in the ground vibrational state (GS) are difficult
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
16/99
16
to determine. Here, vT indicates torsional quanta, k is the eigenvalue of the z-componentof the total angular momentul, and ` is the quantum number associated with the vibrationalangular momentum.
Second, the molecular Hamiltonian is implicitly subjected to contact transformationsby neglecting terms off-diagonal in the Hamiltonian, thus resulting in large model errors
on the leading molecular parameters, such as those characterizing the torsional potential.
The first mechanism is essentially a technical problem whereas the second mechanism pre-
dominantly arises from insufficiency in the vibration-torsion-rotation theory and practical
problems in spectrum analyses.
The first result of the selection rules is that the conventional microwave and millimeter-
wave spectra are insensitive to the leading terms in the torsional Hamiltonian. However,
there are few methods to overcome the complications arising from the electric dipole selec-
tion rules. The selection rules (k = = 0) can be broken by applying molecular-beamelectric-resonance (MBER) spectroscopy [13] which is capable of providing dipole mo-
ment information and accurate measurements of(A0 B0) quantity and the leading tor-sional parameters V0,3 and . The MBER measurements form an excellent starting point formolecular studies when no other information is available. Unfortunately, they are confined
to the lowest rotational states in the vibrational and torsional ground state, and therefore
cannot be used to determine energy levels with a wide range of rotational and torsional
quantum numbers.
A second result of the selection rules is that the pure torsional spectra are forbidden in the
lowest order. However, the torsional transitions with (vT 6= 0,k = 0; = 0)becomeweakly allowed by second-order effects, such as intensity capture through torsion-mediated
interactions mechanisms and, on the other hand, higher terms of the electric dipole moment
expansion [4, 5]. The torsional bands typically fall in the far-infrared region and the linestrengths are only 0.1% of those of infrared active fundamental bands. Therefore, exper-iments on torsional bands require specific experimental conditions, such as high sample
pressure, long absorption path length, low sample temperature and long recording time of
the spectra.
The large amplitude motion in a periodic potential complicates the quantum mechanical
treatment in many respects. The Gaussian basis set is sufficient to characterize small am-
plitude vibrational motion, thus enabling the use of simple and closed-form expression for
eigenenergies, transition frequencies and wave-functions. However, in the case of large am-
plitude motion in the periodic potential, the Gaussian basis set can no longer be used, and
the torsional wave-functions require a plane wave expansion. This leads to computational
challenges and poses problems for analytical approach.
One aspect of this thesis is to extend the energy level consideration to the lowest funda-
mental bands in methyl silanes and to develop the quantum mechanical model to charac-
terize the overall energy levels accurately with a relatively small number of spectroscopic
parameters. This involves basic understanding on mechanisms that couple vibrational de-
grees of freedom with the torsional degree of freedom.
This thesis consists of 6 chapters. Chapter 2 is devoted to the background of the papers
and general circumstances affecting this thesis. Chapter 3 gives a brief introduction to the
general theory including a formulation of the energy expression and symmetry considera-
tion of the vibration-torsion-rotation states. Chapter 4 deals with the principles of spectrum
analysis by accounting for specific aspects due to the internal rotation. Moreover, Chapter
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
17/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
18/99
2 Overview and Background of the Papers
I started my post-graduate studies in early 1994 afterfinishing my Masters degree in OuluUniversity. At that time, my fiance Anne was appointed to a post-doctoral position in
the Department of Geography of the University of British Columbia (UBC), Vancouver,
starting March 1995. I presented the idea of carrying out a part of my post-graduate studies
in Canada to my supervisor Professor Anttila who was compliant to my idea from the
beginning and suggested to contact professor Irving Ozier at the Department of Physics
and Astronomy at the UBC. After some consultations with professor Ozier and financial
arrangements, I started working in professor Oziers laboratory on my internal rotation
projects.
2.1. Paper I
My first task at the UBC was to modify an analysis program for analyses of symmetric
internal rotors to be operable in a UNIX-based system. The analysis program was origi-
nally written by professor Leo Meerts from University of Nijmegen for analysing torsional
spectra in the GS. The program was later modified to treat coupled two-band systems of
the GS and the lowest-lying perpendicular fundamental band 12 of CH3SiH3 [6].
At that time, unpublished data were available for the lowest fundamental band 12 of
deuterated methyl silane CH3SiD3. The purpose of the 12 band study was to examine the
isotopic dependence of the torsional properties between CH3SiD3 and CH3SiH3. The dataset of CH3SiD3 was not as developed as that of CH3SiH3 since the torsional bands 26 and36 6 were not available. However, based on the experience on CH3SiH3 [6], we as-sumed that the vibrational-torsional-rotational interactions between the (v12 = 1) state andthe torsional dark states (v6 = 3) in the GS would provide information on the torsional darkstates so that the correlations between the leading torsional parameters would be avoided,
and thus a more realiable torsional Hamiltonian would be determined than was obtained in
the previous study [2] based on the MBER measurements. After a thorough investigation,
we had to admit that the data were not able to break the correlations between the molecular
parameters to the extent that the second torsional potential term V6, for example, could be
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
19/99
19
obtained with sufficient reliability. Nevertheless, the 12 band was fit within experimental
accuracy and it was analyzed taking into account the torsion-mediated interactions between
the GS and the (v12 = 1) state. The results for CH3SiD3 were published in Paper I in 2000.
I carried out the assignment and analysis of the spectra and wrote thefi
rst draft of the paper.
2.2. Paper II
The next molecule to be investigated was a quasi-spherical top CH3CF3. This took place in
fall 1995. A few years earlier, Irving Ozier had recorded a forbidden rotational Q-branch
with (K+ 3 K) transitions in the (v6 = 0) and (v6 = 1) torsional states in Zurich byusing a Fourier transform microwave spectrometer. The forbidden Q-branch data were
combined with existing MBER measurements [3] and pure rotational transitions taken from
literature [7]. In Paper II, the theoretical questions were addressed in two different areas.
The first question concerned the torsion-rotation Hamiltonian and the second question dealt
with the (A1 A2) splittings occurring in a near-spherical top exhibiting internal rotation.My contribution to Paper II was focused on the analysis of the (A1 A2) splittings. Atthat time, the analysis program was not capable of treating Hamiltonians with terms off-
diagonal in the k quantum number within one vibrational or torsional state, and the analysisprogram had to be modified accordingly. To my knowledge, this was the first time that
(A1 A2) splittings were examined in a symmetric top with internal rotational degree offreedom by accounting for the torsional effects. One purpose of analyzing the (A1 A2)splittings was to find a physically relevant Hamiltonian that characterizes the (A1
A2)
splitting and to test different representations of the interaction Hamiltonian. A carefulanalysis of the (A1 A2) splittings was made, but unfortunately, a torsional dependencein the (A1 A2) splittings was not observed. In this study, I analyzed the spectra andmodified the analysis program to treat the interaction Hamiltonian coupling the rotational
k sub-states.
2.3. Paper III
An important step in the internal rotation studies was taken when Irving Ozier and NasserMoazzen-Ahmadi introduced me high-resolution infrared data on the lowest-frequency par-
allel fundamental band 5 of CH3SiH3 measured a few years earlier in Ottawa. The prelim-
inary analysis of the 5 band showed strong torsional anomalies that could not be explained
in an isolated 5 band scheme. There were assumptions that the anomalies would arise from
torsion-mediated anharmonic-type interactions between the GS and the (v5 = 1) state, andthe basic form of the interaction Hamiltonian was deduced by symmetry. At that time,
the data set of CH3SiH3 included measurements in the lowest vibrational state (v12 = 1)state and the (6 = 0, 1, 2, 3) states in the GS. Furthermore, the basics of the interactionmechanisms between the (v12 = 1) state and the GS were well known.
The analysis program was modified to treat the triad of the GS, the (v12 = 1) state andthe (v5 = 1) state. The analysis of the triad was complicated by several reasons. First, to
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
20/99
20
my knowledge this was the first time that a torsion- mediated anharmonic-type interaction
mechanism was studied in a symmetric top molecule with high resolution, and information
on such mechanisms in the literature was limited. Second, the perturbations in the (v5 = 1)
state did not allow running afi
t for an isolated 5 band so that unperturbed molecularparameters for the (v5 = 1) state would have been obtained. On the other hand, the GSwas perturbed by the (v12 = 1) state so that it was impossible to analyze separately thedyad of the GS and the (v5 = 1) state. Consequently, the 5 band had to be analyzed witha three-band model from the beginning.
The analysis program at that time treated the quantum mechanical problem in two suc-
cessive stages. In the first stage, the torsional problem was solved separately for the GS,
the (v12 = 1) state and the (v5 = 1) state , thus resulting in a first- order effective torsionalHamiltonian. In the second stage, the vibrational-rotational problem including the vibra-
tionally off-diagonal terms was solved by using the first-order effective torsional Hamil-
tonian obtained from the first stage. However, the second stage neglected second-order
terms on the basis of perturbation theory. It should be noted that a corresponding approx-imation was found relevant in the GS/12 study of CH3SiH3 [6] and CH3SiD3 of Paper
I, and there was no reason to question the relevance of the approximation. We suspected
that the approximation might cause a problem, but it was impossible to separate between
an approximation problem and unknown details of the possible interaction mechanisms
coupling the GS and the (v5 = 1) state. Finally, the analysis program was partly rewrittenso that the full Hamiltonian was solved exactly in one step without using the perturbation
theory. The exact treatment revealed a severe limitation caused by the perturbation theory
in the treatment of the torsion-mediated anharmonic-type coupling between the GS and the
(v5 = 1) state in CH3SiH3.
The results from the 5 study were surprising. The analysis showed that the torsionalHamiltonian in the GS is substantially modified when the torsion-mediated anharmonic-
type effects between the (v5 = 1) state and the GS are treated explicitly. For example, thesecond term V6 in the effective torsional potential in the GS turned out to be an artifactdue to the torsion-mediated Fermi-type interaction mechanisms. Second, the strongest
coupling between the GS and the (v5 = 1) state can occur even if the perturbing torsionalenergy level is well above the torsional barrier, the torsional quanta differing by 5 units
between the two interacting states. This result reveals that a vibrational-torsional-rotational
energy manifold becomes practically chaotic when the higher energy levels systems are
considered. In Paper III, I was responsible for the assignment of the 5 spectrum, theanalysis of the data, and writing the first draft of the paper.
2.4. Paper IV
Paper I showed that the molecular beam measurements from ref. [2] and the high resolution
infrared measurements in the 12 band of CH3SiD3 were not capable of enabling a stringest
test to be made for the torsional Hamiltonian. After returning back to Finland from the UBC
in spring 1997, a project was launched to record the 26 and 36 6 torsional bands inCH3SiD3 with high resolution Fourier transform infrared (FTIR) spectroscopy.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
21/99
21
Recording pure torsional spectra is a challenging task because the spectral transitions
are weak and they fall in the far-infrared region. Our research group had initiated a collab-oration with MAX-I electron storage ring in Lund, Sweden, where they have an interfer-
ometer coupled to an infrared beam line that acts as a light source for the interferometer. Inan ideal case, an electron storage ring light source can provide considerably more intense
radiation than the thermal sources traditionally used in infrared spectroscopy, thus reducing
time needed for recording the spectra. Furthermore, a coolable multipass absorption cell
attached to the spectrometer was available, thus enabling to cool the CH3SiD3 sample in
order to increase the intensity of the torsional spectra.
The 26 and 36 6 spectra were recorded with 0.002 cm1 resolution by using theelectron storage ring during a period of one week in spring 1997. Furthermore, the 12spectrum was remeasured in Oulu at an improved resolution of0.00125 cm1 in order toobserve the torsional splittings in the (v12 = 1) state, to increase the rotational quantumnumber range, and to observe the hot band 12 + 6
6.
The primary results of the GS/12 study in Paper IV were twofold. First, it was shownthat an electron storage ring may be applied for a high resolution far infrared measurements
when a relatively long recording time and a long absorption path length are required. Sec-
ond, the analysis showed that the interaction between the GS and the (v12 = 1) state canbe characterized with two models that have different physical interpretation. In Paper IV, I
took the initiative to carry out the experiments and recorded the torsional spectra with Dr.
Veli-Matti Horneman. Furthermore, I carried out the spectrum analysis and wrote the first
draft of Paper IV.
2.5. Remarks on the Studies
The development in understanding the torsional interaction mechanisms dealt in Papers I,
II, III and IV is chrystallized in the capabilities of the analysis program. To my knowl-
edge, there are no program packages available that are capable of carrying high resolution
multi-band analysis of symmetric tops with internal rotation. Therefore, testing of mod-
els requires continuous calculational development and implementation of new Hamiltonian
terms into the analysis program. Furthermore, a great deal of effort was put to improve the
efficiency, flexibility and usability of the analysis program. This is not directly indicated in
scientifi
c publications, even though it has a substantial effect on the quality of the studies.From a personal point of view, internal rotation in symmetric tops provides a wide range
of motivating scientific challenges. The torsional motion as such provides an isolated and
well-defined problem even in relatively large molecules, and the analyses result in intu-
itively understandable molecular parameters, such as the torsional barrier height.
Furthermore, the torsional interaction mechanisms of symmetric tops are relatively un-
known, and new fundamental information may be generated quite easily from the analyses.
Results from Paper III may be taken as an example in this respect.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
22/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
23/99
23
3. Derive a classical Hamiltonian, which involves further specification of the molecular
coordinate system.
4. Obtain a Hamiltonian operator from the classical Hamiltonian.
5. Expand the Hamiltonian operator in respect to vibrational and torsional variables.6. Define the zeroth order Hamiltonian operators for the vibrational, torsional and rota-
tional degrees of freedom. Solve a Schrdinger equation for each one to obtain basis
functions for the full Hamiltonian operator.
7. Apply symmetry properties of the molecule to simplify the Hamiltonian operator and
the basis set.
8. Obtain an effective Hamiltonian by solving the Schrdinger equations for the vibrational-
torsional-rotational Hamiltonian.
3.1. Born-Oppenheimer Approximation
The spin-free rovibronic Schrdinger equation including the nuclei and electrons is [12](~
2
2
lXr=2
2rmr
+~2
2M
lXr,s=2
r s +lX
r
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
24/99
24
time scale, only depend on the nuclear positions. In terms of perturbation theory, the Born-
Oppenheimer approximation can be applied, when a separation of two electronic states that
are coupled by a vibrational-torsional-rotational-electronic mechanism is large compared
to the coupling matrix element, and the effects from the approximation are smaller thanthe accuracy of the measurements. This assumption is easily fulfilled as the electronic
ground state is well below the excited electronic states, and the rovibrational consideration
is confined to the lowest vibrational states.
The original form of the Born-Oppenheimer approximation involved separation of the
nuclear and electronic motions only. With a similar analog, the molecular wave-function
can be separated into an electronic part, a vibrational part, a torsional part and a rotational
part [16]. The coordinates yielding such a separation are referred to as Born-Oppenheimer
variables. In the following discussion, the electronic variables are neglected and the con-
sideration is confined to the ground electronic state.
3.2. Coordinate system for a Symmetric Top with Two Three-Fold
Coaxial Rotors
In the expression of the Born-Oppenheimer variables in a symmetric top molecule with
two three-fold coaxial rotors, five Cartesian coordinate systems are needed. They are as
follows:
1. Space-fixed axis system (X0, Y0, Z0) . Laboratory coordinate system.
2. Center-of-mass-fixed axis system (X,Y,Z) . Parallel to the (X0
, Y0
, Z0
) system. OriginR0 = (X0, Y0, Z0) fixed to the center of mass of the molecule.
3. Molecule-fixed axis system (x,y,z) . Origin (x0, y0, z0) coincides withR0. The z-axiscoincides with the molecular symmetry axis in the reference configuration. Related to
the (X,Y,Z) system by the Euler angles , and . See Fig. 1 for illustration.
4. Top-fixed axis system (xT, yT, zT). The z-axis is parallel to the molecular symmetryaxis. The top-fixed axis system is fixed to one of the coaxial rotors.
5. Frame-fixed axis system (xF, yF, zF). The z-axis is parallel to the molecular symmetryaxis. The frame-fixed axis system is fixed to the other coaxial rotor.
An instantaneous position vector of the nucleus i of a molecule in the space-fixed axis
system is given by a vector
Ri = R0 + S1 (, , ) ti, (2)
where
S1 (, , ) =
cos sin 0 sin cos 0
0 0 1
cos 0 sin 0 1 0
sin 0 cos
cos sin 0 sin cos 0
0 0 1
(3)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
25/99
25
(1-)
X
Y
Z
x'
N
z
x''
x
y
O
a) b)
1
2 3
46
5
xT
xF
yT
yF
c)
Fig. 1. (a) Definition of the Euler angles. The successive rotations of the
Euler angles , and generate the following coordinate transformation:
(X , Y , Z ) (x0, N , Z ) (x00, N , z) (x , y , z) . (b) Definition of the frame-fixed and
the top-fixed coordinate systems and the numbering scheme of nuclei in a methyl silane-like
molecule. The z-axis points from the frame to the top. (c) Definition of the torsional angle .
The frame is displaced by from the reference configuration (solid lines) whereas the top is
displaced by (1 ) to the opposite direction. The total torsional displacement between the
top and the frame is .
is the inverse of the direction cosine matrix [17]. It relates the (x,y,z) axis system to the(X,Y,Z) axis system. The instantaneous position of nucleus i in the (x,y,z) system isgiven by the vector
ti = S1i (0, 0, r) (ai + di) , (4)
where ai and di represent reference positions and the vibrational displacements, respec-
tively, of the top and the frame nuclei in the top and frame-fixed axis systems (T/F system).
The transformation from the T/F system to the (x,y,z) system is given by
S1i (0, 0, r) =
cos r sin r 0 sin r cos r 0
0 0 1
i frame; r =
i top; r = 1 (5)
which rotates the top by angle relative to the frame and then the entire molecule by about the z-axis in the (x,y,z) axis system, see Fig. 1 (b) for illustration.
The S1i (0, 0, r) matrix can explicitly be included in the S1 (, , ) matrix by re-
defining the Euler angles for the top and the frame separately. It is easy to show that the
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
26/99
26
Euler angles transform to
F = T = (6)
F = T = (7)F = (8)T = + (1 ) (9)
T F = , (10)
where subscript T refers to the top and F to the frame, respectively. Parameter can be anarbitrary number assuming values between 0 and 12 . Specific values of parameter will bediscussed in Section 3.5.
3.3. Classical Kinetic Energy of a Rotating and Vibrating Molecule
In the adopted coordinate system, the total kinetic energy ofN mass points of a moleculecan be written as
2TTOT =NX
i=1
mifRiRi
= fR0R0N
Xi=1 mi trans+ 2 gSR0SS1 NX
i=1
miti +NX
i=1
miti
!trans-rot-tor
+NX
i=1
mi
^SS
1ti
SS
1ti
rot, rot-tor
+ 2NX
i=1
mi
^SS
1ti
ti vib-tor-rot
+NX
i=1
mi etiti, vib, tor, vib-tor (11a)where the matrix notation involving factorSS
1can be replaced with
SS1v = v, (12)
where v = (vx, vy, vz) is an arbitrary vector. Expecially,
Si (0, 0, r) S1 (0, 0, r)v =rk v, (13)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
27/99
27
where k is a unit vector along the molecularz-axis. The components of the angular velocityvector in the (x,y,z) system can be obtained from the associated angles and can bewritten as
x = sin sin cos (14)y = cos + sin sin (15)
z = cos + (16)
= (17)
Quantity defines the relative angular velocity of the top and the frame about the molec-ularz-axis.
The left hand side of coordinate transformation (2) contains 3 translational, 3 rotational,1 torsional, and 3N 7 vibrational degrees of freedom and therefore, 3N independent co-ordinates are required to express the total kinetic energy. However, on the right hand sideof the kinetic energy expression (11a) there are 3N + 7 coordinates in total: there are 3translational coordinates (X0, Y0, Z0, ), 3 angular coordinates (, , ) , 1 torsional coor-dinate and 3N vibrational displacement coordinates. Therefore, 7 constraint equationsare required to orthogonalize the Born-Oppenheimer variables. They can be obtained from
equations
NXi=1
miS1 (0, 0, r)di = 0 (18)
NXi
miS1
i (0, 0, r) (ai di) = 0 (19)
k
"XTF
mi (ai di)
#= 0. (20)
Eq. (18), also called the first Eckart condition [18], provides 3 constraint equations andconserves the center of mass during a simultaneous vibrational and torsional displacement.
Eq. (19) is called the second Eckart condition and provides 3 more constraint equationsthat remove the redundancy between the vibrational displacement coordinates and the Euler
angles. Furthermore, the molecule-fixed coordinate system is redefined so that the angu-
lar momentum arising from the vibrational motion is minimized. The term (20) called aSayvetz condition [19], is a large amplitude analog to the second Eckart condition and pro-
vides one constraint equation that removes the redundancy between the torsional angle and
the vibrational displacements. It also redefines the frame-fixed and top-fixed axis systems
so as to minimize the interaction between the vibrational angular momentum and torsion.
In Eq. (20), the abbreviationP
TF in summation isXTF
Di = (1 )XiT
Di XiF
Di,
where T and F correspond to summation over top and frame atoms, respectively.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
28/99
28
The constraint equations (18)(20) along with omission of the first term in Eq. (11a)
eliminate the translational contribution from kinetic energy. After applying Eq. (12) and
the constraint equations (18)(19), the kinetic energy expression becomes
2TV TR =3N7Xi=1
mi di di vib (21a)
+X
,=x,y,z
I rot (21b)
+2
h(1 )2 ITzz + 2IFzz
itor (21c)
+2
NXi=1
miS1
i (0, 0, r) hdi dii vib-rot (21d)+2
X=x,y,z
(1 ) ITz IFz
tor-rot (21e)
+2k
"XTF
mi
hdi di
i#vib-tor (21f)
+2 (k)XTF
miS1i (0, 0, r) (ai di)
, vib-tor-rot (21g)
where IT and IF are inertial tensor components of the top and the frame, respectively,
and I is an inertial tensor component of the entire molecule in such a way that I = IT
+IF. Furthermore, for the reference configuration,
IT,e = IF,e = I
e = 0 6= = x,y,z (22)
at any torsional angle . A kinetic energy expression similar to (21a)(21g) has also been
given in Ref. [10].Let us assume that the torsional motion is frozen i.e., = = 0. Then, terms(21a), (21b) and (21d) retain, and the kinetic energy expression (21a)(21g) reduces to
that of a semi-rigid symmetric rotor molecule: term (21a) arises from the small amplitude
vibrational motion, term (21b) represents the overall rotation of the molecule, and term
(21d) represents the coupling of the overall rotation and the small amplitude vibration.
The additional terms are due to internal rotation. Term (21c) represents the pure torsional
kinetic energy due to relative angular motion of the two coaxial rotors about the molecular
z-axis. Term (21e) represents the Coriolis interaction energy between overall rotation andtorsion. Term (21f) is similar to (21d): it arises from the coupling between the vibrational
and torsional angular momenta.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
29/99
29
Term (21g) arises from the fact that the time derivative of the second Eckart equation
(19) is not simply
PNi mi
hS1i (0, 0, r)
ai di
ibut
d
dt
" NXi
miS1i (0, 0, r) (ai di)
#=
NXi
mihS1i (0, 0, r)
ai di
i+k
XTF
miS1i (0, 0, r) (ai di)
= 0, (23)
which implies that the Eckart conditions cannot perfectly be fulfilled in the case of large
amplitude motion. Term (23) would vanish if the second Eckart condition (19) held for
both the top and the frame separately.
The components di, ( = x,y,z) and their time derivatives di,k of the cartesian dis-placement coordinates di occurring in the kinetic energy expression (21a)(21g) can be
written in terms of vibrational normal coordinates Qk as
di,k =NX
k=1
m1/2i li,kQk (24)
di,k =NX
k=1
m1/2i
hrl
0i,kQk + li,kQk
i, (25)
where li,k = li,k (r) is in general a function of the torsional angle implying the torsionaldependence of the vibrational motion of a molecule. Furthermore,
l0i,k =li,k (r)
. (26)
Note, that the li,k constants are defined in the T/F systems for the top nuclei and theframe nuclei, respectively. By treating the torsion formally as a fourth rotational degree of
freedom and adopting the normal coordinate presentation for the vibrational variables, the
kinetic energy can be written in the conventional form [8] as
2TV TR =3N7Xk=1
Qk +
X=x,y,z,,l
X,l
lkQl
2
+X
,=x,y,z,
I0. (27a)
The form of classical kinetic energy (27a) is identical to that of a semi-rigid rotor with
additional summation overin the rotational part. However, the components of the inertial
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
30/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
31/99
31
Table 2. Definitions of the constants
kl = ,Tkl +
,Fkl = x,y,z
kl = 0
kl + (1 ) z,Tkl z,Fkl,Mkl =
P=x,y,z
S1 (0, 0, r) ,Mkl
a,b = x,y,z
,Mkl =
PiM
(li,k () li,l () li,k () li,l ()) c 6= 6=
0
kl =P
i,=x,y,zli,k () l
0i,l ()
d
a r= for M=F and r=1- for M=T.b S1 (0, 0, r) is the matrix element of the S
1 (0, 0, r) matrix.c Summation over the frame (M=F) or the top (M=T) atoms, respectively.d Index i runs over all the nuclear coordinates if not otherwise specified.
3.4. Watson Hamiltonian
When the vibrational amplitudes are small compared to the interatomic distances, the Tay-
lor expansions of the inverse of the tensor and that of the potential energy V (,q)usually converge with a rapidity justifying the use of the perturbation methods. Further-
more, James Watson [20] has shown that terms in the expansion of tensor can bepresented with the components of the inertial tensorIe at the equilibrium and with its firstderivatives with respect to normal coordinates. This leads to a relatively simple form of the
vibrational-torsional-rotational Hamiltonian, also called the Watson Hamiltonian, that can
be expressed in wavenumber units as
H =1
2
X,=,y,z,
B (J p) (J p) + 12
3N7Xk=1
kp2k +V (,q) + U, (32)
where B = hc and k is a harmonic wavenumber of normal mode k. The units are[B] =[k] =cm
1, [c] =cm/s, and [V (,q)] =cm1. The term U acts as a mass depen-dent contribution to the potential energyV (,q). The dimensionless normal coordinatesqk, their conjugate momenta pk, and the vibrational angular momentum are defined as
qk = k~21/4Qk = 1/2k Qk (33)
pk i
qk(34)
p =3N7X
kl
kl
l
k
qkpl, (35)
where k = 42k where k is a harmonic frequency of a normal mode k in frequency units.
When a molecule exhibits a large amplitude motion, the commutation relations between
the B tensor and the molecular variables required by the Watson Hamiltonian are not
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
32/99
32
Tabl e 3. Definitions of the constants
k=
k= (
,x
,y)
T
PM=F P=x,y,z rS1 (0, 0, r) ,Mk a = x, yk =
k =
k =
k =
zk =
zk = 0 , = x,y,z
,Mk =
PiM
m1/2i (aili,k aili,k) b 6= 6=
a See footnotes a and b in Table 2.b See footnote c in Table 2.
necessarily fulfilled. This is due to the -dependence of the B tensor. Furthermore, thecommutation relations depend on the choice of the internal coordinate system i.e., on the
value of the parameter which is not yet specified. For the simplicity of the followingconsideration, the Watson form of the Hamiltonian will be used as a starting point. An
anticommutator form of the operators is used when required.
3.4.1. Expansion of the Watson Hamiltonian
The separation of the normal vibrations and internal rotation may be considered from a
viewpoint similar to that adopted in the Born-Oppenheimer separation of electronic and
nuclear motions: the time period of the internal rotation is much larger than that of thenormal vibrations, and the vibrational constants depend parametrically on the torsional
angle. As a consequence, the B tensor may be written as
B () = B0 () +
3N7Xk
Bk ()qk +1
2
3N7Xkl
Bkl ()qkql, (36)
where the coefficients in the Taylor expansion are
B0 =
~2
2hc
1
Ie
!(37)
Bk =
~2
2hc
qk
0
(38)
Bkl =
~2
2hc
2qkql
!0
. (39)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
33/99
33
Table 4. Definitions of the constants
kl=
kl=
F
PM=T P=x,y,z rS1i (0, 0, r) ,Mkl a = x,y,zkl =
zkl +
12k
0kl
kl =
kl = 0
zkl= (1 ) z,Tkl z,Fkl,Mkl =
PiM
hli,k () l0i,l () li,k () l0i,l ()
i, , = x,y,z
6= 6=
k0kl=Pi,
l0i,k () l0i,l ()
b
a See footnotes a and b in Table 2.b Index i runs over nuclear coordinates if not otherwise spesified.
The coefficients B0 , Bk and B
kl can be expanded in Fourier series as
B0 () = B0,0 +
Xn=1
B,c0,n (1 cos3n) + B,s0,n sin3n (40)
Bk () = Bk,0 +
Xn=1
B,ck,n (1 cos3n) + B,sk,n sin3n (41)
B
kl () = B
kl,0 +
Xn=1
B,c
kl,n (1 cos3n) + B,s
kl,n sin3n. (42)
In the B,l,m notation, indices and run over coordinates x,y,z and . The super-scripts c and s indicate cosine and sine series expansion, respectively. The subscipt l in-dicates the normal coordinate, and subscript n indicates an order in the Fourier series ex-
pansion. In the case = 0, the expansion of the quantity B reduces similar to that of a
semi-rigid rotor: B0,0 are the rotational and torsional constants in equilibrium, while Bk
and Bkl,0 are the rotational and torsional derivatives.
3.4.2. Potential Function
A normal coordinate analysis of a molecule exhibiting small amplitude vibrations about
a single equilibrium configuration can be performed according to the standard GF ma-
trix method [17]. However, for molecules with a large amplitude torsional motion, the
standard treatment cannot be applied. In the GF matrix formalism, provided that the Born-
Oppenheimer approximation is valid, the kinematic matrix is independent on the torsional
angle. However, the elements of the force constant matrix possess torsional dependence
and therefore, the vibrational problem of the molecule has to be solved separately for each
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
34/99
34
torsional angle [21]. On the other hand, the geometry of the molecule is a function of the
normal coordinates, and hence the barrier to internal rotation depends on the vibrational
state. Therefore, in a general case, the Born-Oppenheimer potential function cannot be
fully separated into the vibrational and torsional part. Let us assume that the classical forcefield calculation in an arbitrary coordinate system has been carried out by taking into ac-
count the torsional effects on the initial force constant matrix. In such a case, the potential
function can be expressed as
V (, q) =3N7X
k
k ()qk +1
2
3N7Xk
()q2k +1
6
3N7Xklm
kklm ()qkqlqm (43a)
+
Xi1
2V3i (q) (1 cos3i) . (43b)
The first three terms make up the small amplitude vibrational potential having an implicit
0.0
0.1
0.2
0
12
3
0
200
400
600
0 1 2
0
200400
600
compo
nenso
eorsonapo
en
a
ncm
V6
V9
V3
TOTAL
Fig. 2. The Fourier components of the torsional potential of CH3SiD3. See Eq. (43b) for the
formula. The torsional parameters are obtained from Paper IV. The values are: V3 = 583.9403
cm1, V6 = 2.7858 cm1 and V9 = 0.1878 cm
1 .
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
35/99
35
torsional dependence. The torsional potential (43b) is given in Fourier series with constants
V3, V6, V9, . . , . The formation of the total torsional potential from its Fourier componentsin CH3SiD3 is exemplified in Fig. 2.
The small amplitude force constants k () , k () , and kklm () can be expanded inFourier series as
k () = k,0 +X
n=1
ck,n (1 cos3n) +Xn
sk,n sin3n (44)
k () = k,0 +X
n=1
ck,n (1 cos3n) +Xn
sk,n sin3n (45)
kklm () = kklm,0 +X
n=1
kcklm,n (1 cos3n) +Xn
ksklm,n sin3n, (46)
where the cosine and sine terms are indicated with superscripts as before. When 0, thevibrational potential converges to that of a semi-rigid rotor. Therefore, it is required that
k,0 = 0, and constants k,0 and kklm,0 are the force constants of a semi-rigid molecule.The torsional potential V3i (q) can be expanded in Taylor series in terms of the normal
coordinates as
V3i (q) = Ve3i +
3N7Xk
Vk3iqk +1
2
3N7Xkl
Vkl3i qkql+..., (47)
where Ve3i is the torsional potential expansion coefficient in the equilibrium configuration,
and Vk
3i and Vkl
3i are derivatives of the components of the torsional barrier expansion withrespect to normal coordinates.
When terms (44)(46) are substituted to the potential function (43a)(43b), and terms
with identical torsional and vibrational operators are collected together, the following in-
separable coefficients are obtained:
Vk3i = Vk3i + 2
ck,i (48)
Vkk3i = Vkk3i +
ck,i. (49)
It is obvious that components Vk3i and 2ck,i and components V
kk3i +
ck,i, respectively,
cannot be separated from each other by spectroscopic methods.
The torsional dependence of the GF problem leads to torsional dependent , and constants given in Tables 24. It is customary to express them as Fourier series
kl = kl,0 +
Xn=1
h,ckl,n (1 cos3n) + ,skl,n sin3n
i; (50)
k =
k,0 +
Xn=1
h,ck,n (1 cos3n) + ,sk,n sin3n
i; (51)
kl =
kl,0 +
Xn=1h,ckl,n (1 cos3n) + ,skl,n sin3ni
. (52)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
36/99
36
At the infinite barrier limit 0, kl kl,0 ( = x,y,z), which corresponds to thecase of a semi-rigid molecule. From the definition of
kl given in Table 2 one can deduce
that kl (1 ) z,Tkl z,Fkl by assuming that l0i,k ( = 0) = 0. Also kl 0.
The consideration of the k constants is more complicated as they arise from the specificEckart condition given by Eq. (23).
The Hamiltonian resulting from substituting the expansions ofB and V (,q) intothe Watson Hamiltonian (32) consists of terms of type M(V , T , R) VkTmRl, whereVk, Tm, and Rl are the vibrational, torsional, and rotational operators, respectively, andM(V , T , R) is the associated molecular constant. Indices k, m, and l are the degreesof the operators. In this thesis, we follow the approach introduced by Aliev and Watson
[22] and later modified for the internal rotor problem by Duan and co-authors [13]. In an
Hlm notation, l represents the degree of the vibrational operators qk and pk, whereas mrepresents the degree of the torsional and rotational operators. In this case, the notation of
the torsional angular momentum follows that conventionally applied to the overall angularmomenta. However, terms with (1 cos3n) and sin3nare classified according to m =2n. For example, the degree of term with cos3 is 2.
In this notation, the lowest order molecular Hamiltonian can be expressed as
H = H20 +H30 +H40 + .. vib (53a)
+H12 +H21 +H22 + .. vib-tor-rot (53b)
+H02 +H04 +H06 + .. tor+tor (53c)
The lowest order terms of the Hamiltonian H are tabulated in Tables 5 and 6. The op-
erators are not written in the Hermitian form since the commutation relationships betweenthe angular momentum operators are yet not known due to the unspecified parameter.Furthermore, the torsional angular momentum operatorJ does not commute with cos3nand sin3n. When the operators are written explicitly, an anticommutator form has to beused.
TermsH20,H30, andH40 constitute the pure vibrational terms, theH20 representing anisolated harmonic oscillator, and H30 andH40 being the cubic and quartic anharmonic po-
tential terms, respectively. TermH12 contains the conventional centrifugal distortion terms
of a semi-rigid rotor with additional torsional contribution. Furthermore,H12 contains an
anharmonic-type term with qk (1 cos3) arising from the linear vibrational dependenceof the torsional barrier height plus the torsional dependence of the linear potential term.Term H21 contains the conventional rigid rotor-harmonic oscillator Coriolis terms with
additional torsional contribution. Term H22 describes the quadratic stretching of the iner-
tial tensor and the quadratic change of the torsional barrier as a function of the vibrational
coordinate. TermsH31 andH14 are higher order corrections to H21 andH12, respectively.Terms in H02 make up the rigid rotor Hamiltonian including the zeroth order kinetic
torsional term and the cross terms between the torsion and overall rotation. Furthermore,
the first Fourier term in the torsional potential expansion is included. The quartic term
H04 and the sextic term H06 are characteristic to an internal rotor molecule only: they
characterize the torsional dependence of the rotational constants and also include higher
terms in the torsional potential expansion.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
37/99
37
Table 5. Terms in the vibrational-torsional-rotation Hamiltoniana,b
Vibrational terms
H20 =12Pk k q2k + p2k
H30 =16
Pklm
kklmqkqlqm
H40 =124
Pklmn
kklmnqkqlqmqn +P
B0,0p2
Vibrational-Torsional-Rotational termsH12 =
Pkl
Rkqk
H21 =
Pkl
Rlkqkpl
H22
=Pkl RklqkqlH31 =
Pklm
Rmklqkqlpm
H14 =Pkl
Tkqk
Torsional-Rotational terms
H02 =P
B0,0JJ +12V
e3 (1 cos3)
H04 =
P
hB,c0,1 (1 cos3) + B,s0,1 sin3
iJJ +
Ve6
2 (1 cos6)
H06 =PhB,c0,2 (1 cos6) + B,s0,2 sin6iJJ + Ve92 (1 cos6)
a Only terms relevant to this context are considered.b For a general consideration, a Hermitian form of operators is required.
3.4.3. Rigid Rotor-Symmetric Top
In the treatment of the rotational problem of a polyatomic molecule, the rigid rotor-symmetric
top model is most commonly used as a zeroth order of approximation. The rigid rotor-
symmetric top Hamiltonian arises from operatorH02 given in Table 5 and can be writtenas
H02,r = Bxx0,0J
2 +
Bzz0,0 Bxx0,0J2z. (54)
The rotational constants have been defined in Eq. (37) and J2 = J2x + J2y + J
2z. The
dimensionless angular momentum operators can be expressed as functions of Euler angles
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
38/99
38
Table 6. Torsion-rotation operatorsa
Rk =P Bk,0JJ +
12
Vk3i (1
cos3) b
Rlk =2ql
k
P
B0,0klJ
Rkl =12
"P
Bkl,0JJ +12
Vkl3 (1 cos3)# b
Rmkl =2q
mkl
P
Bk,0klJ
Tk =P
hB,c1,1 (1 cos3) + B,s1,1 sin
iJJ +
Vk6
2 (1 cos6)a For a general consideration, an anticommutator form of operators is required
b Vk3 = Vk3 + 2ck,0; Vkl3 = Vkl3 + klck,1as
Jx = i 1
sin cos
+ sin
+ cot cos
(55)
Jy = i
1
sin sin
+ cos
cot sin
(56)
Jz = i
, (57)
which satisfy the commutation relations
[Jx,Jy] = iJz; [Jy,Jz] = iJx; [Jz,Jx] = iJy. (58)
The molecularz-axis is chosen as a projection axis by requiringJ2,Jz
= 0.
The Schrdinger equation of a symmetric top can be obtained by substituting the angu-
lar momentum operators (55)(57) to the symmetric top equation (54). We obtain the
Schrdinger equation1
sin
sin
+
1
sin2
2
2+
cos2
sin2 +
Bzz00Bxx00
2
2
2cos sin2
2
+
E0rBxx00
0r = 0,
where
E0r (J,k,m) = Bxx00 J(J+ 1) + (B
zz00
Bxx00 ) k
2 (59)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
39/99
39
is the symmetric top eigenvalue. The rotational eigenfunctions are
0r (, , ) = SJkm (, ) e
ik, (60)
where SJkm (, ) = Jkm () eim, where Jkm () is the Legendre polynomial. The
rotational eigenfunctions are fully characterized by the three rotational quantum numbers
J, k and m. The quantum numberJ is associated with the total angular momentum opera-tor obeying a relation
J2 |J,k,mi = J(J + 1) |J,k,mi , (61)
where Diracs notation is used and Jassumes values J = 0, 1, 2,.., . The quantum num-berk assumes values k = J,J+ 1,.., 0, 1, . . ,Jbeing associated with the component ofthe total angular momentum along the molecular axis and satisfying a Schrdinger equation
Jz |J,k,mi = k |J,k,mi . (62)
The magnetic quantum numberm assumes values m = J,J + 1,.., 0, 1, . . ,Jobeyingthe Schrdinger equation
JZ |J,k,mi = m |J,k,mi , (63)
where JZ is the z-component of the total angular momentum in the space fixed coordinatesystem. Since no external fields exist and hence the orientation of the (X,Y,Z) coordinate
system is arbitrary with respect to the molecule, the m quantum number will be droppedfrom the notation in the following consideration.
The matrix elements of the angular momentum operators are fully defined by the com-
mutation relations (58). Let us define a ladder operatorJ followingly:
J = Jx iJy. (64)
The matrix elements of the operatorJ are
hJ, k 1|J |J, ki = pJ(J+ 1) k (k 1), (65)
which can be used to calculate matrix elements for any (J)m
type operator.
3.4.4. Harmonic Oscillator
The harmonic oscillator approximation is the most commonly used method to treat the
small amplitude vibrational problem as it enables a full separation of the vibrational modes
and a simple procedure to obtain the matrix elements. The harmonic oscillator Hamiltonian
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
40/99
40
is given by operatorH20 in Table 5 and yields the Schrdinger equation
1
2
3N7
Xk k p2k + q2k0vk (qk) =3N7
Xk E0vk0vk (qk) . (66)As a result of solving Eq. (66), we obtain the wave-functions and eigenvalues
0v (q1,q2, ..q3N7) =
3N7Yk=1
0vk (qk) (67)
E0v = Ev =3N7Xk=1
k
vk +
dk2
, (68)
where vk
(= 0, 1, 2,..,
) is the vibrational quantum number and dk
is the degeneracy,
respectively, of normal mode k. The vibrational ladder operatorRk is defined as
Rk =1
2(pk iqk) (69)
which operates on the Diracs ket-vector|vki as follows:
Rk |vki = i
svk +
1
2
1
2
|vk 1i . (70)
The vibrational variables satisfy their usual commutation relations
[pk,qk] = [qk,pk] = i (71)[pmk ,p
nl ] = [q
mk ,q
nl ] = 0 m, n = 0, 1, 2, 3.. (72)
The analytical form of the one-dimensional oscillator wave-function is of no interest here.
However, the two-dimensional oscillator wave-function is
vt,l (%, ) = Nvt,le2/2%|lt|L|lt|n (%) e
ilt, (73)
where Nvt,lt is the normalization factor and functions L|l|n () are the associated Laguerre
polynomials. The polar coordinates % and are defined in terms of the degenerate pair ofnormal coordinates as
%2 = q2a + q2b (74)
= arctan
qaqb
. (75)
In Eq. (73), index lt (= vt, vt + 2, . . ,vt 2, vt) is the quantum number which deter-mines the eigenvalue of the vibrational angular momentum about the molecularz-axis. Inthe first excited state vt = 1, there are two states |vt = 1, lt = +1i and |vt = 1, lt =
1i
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
41/99
41
that are associated with degenerate zeroth order energies. However, the degenerary is bro-
ken by a vibrational angular momentum operator defined as
= i
= qtaptb qtbpta, (76)
giving rise to Schrdinger equation
|vt, `ti = lt |vt, lti . (77)
The ladder operators in the two-dimensional case are
qt = qta iqtb (78)
pt = pta iptb, (79)
where the phase convention is chosen so that the matrix elements ofqt are real, and those
ofpt are imaginary. The matrix elements ofqt and pt can be found from Ref. [8].
In the case of a symmetric top molecule with internal rotation, the definition of the
normal coordinates in terms of internal displacement coordinates, Cartesian displacement
coordinates or symmetric coordinates is problematic because the vibrational force field
depends on the torsional angle. To simplify the treatment of the vibrations, the degenerate
vibrations have to be localized either in the top or in the frame. Therefore, the polar angle characterizing the angular motion of a two-dimensional oscillator measured in an arbitrary
(x,y,z) coordinate system is
m = M + r, (80)
where M = F for the frame and M = T for the top. Furthermore, the components of thenormal coordinates and the conjugate momenta transform as
qta,m,M = qta,M cos r qtb,M sin r (81)pta,m,M = pta,M cos r ptb,M sin r (82)qtb,m,M = qtb,M cos r+ qta,M sin r (83)
ptb,m,M = ptb,M cos r+ pta,M sin r, (84)
where r = for the frame and r = (1 ) for the top. It is easy to show that the vibra-tional angular momentum (76) is invariant in the transformation (81)(84). Furthermore,
the ladder operators (78) and (79) transform to
qk,m = qkeir (85)
pk,m = pkeir. (86)
In PAM ( = 0), the treatment of a vibration localized in the frame is similar to that of asemi-rigid rotor.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
42/99
42
3.5. Internal Rotor in Hindering Potential
In the discussion of the classical kinetic energy, the molecular variables are defined in
the (x,y,z) system with an arbitrary orientation and angular velocity with respect to themolecule. The angular velocities in the floating (x,y,z) system are related to those pre-sented in the molecule-fixed axis system with transformation
x,my,mz,m,m
=
cos r sin r 0 0sin r cos r 0 0
0 0 1 r0 0 0 1
xyz
, (87)
where the subscript m indicates a molecule-fi
xed variable. The molecular frame of refer-ence is fixed to the frame when r = and to the top, respectively, when r = (1 ) .It is customary to choose the frame-fixed axis system as a molecular frame of reference
i.e., r = [9]. The Euler angles and are insensitive to the coordinate transformation.However, the nodal Euler angles F, T, and m for the frame, top and for the entiremolecule, respectively, transform according to relations
F = m (88)T = m + (1 ) (89)
m= (1
)
F+
T. (90)
By neglecting the vibrational contribution to the inertial tensor, Eqs. (30) and (87) yield
the components of the angular momenta
Jx,m = Ie,mxx x,m = I
exx (x cos + y sin ) (91)
Jy,m = Ie,myy y,m = I
eyy (x cos y sin ) (92)
Jz,m = Iezzz,m + I
ez,m
= Iezz (z + ) + [(1
)
(1
) ] Iezz (93)
J,m = Ie,m + I
e,mz z,m
=
2 + 2 Iezz + [(1 ) (1 ) ] Iezz (z + ) . (94)In Eqs. (93) and (94), the constant is defined as a ratio of the inertial moment of the top
and that of the entire molecule, that is
Ie,Tzz
Iezz. (95)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
43/99
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
44/99
44
to
HP02 = Bxx00J
2 +
Bzz,F00 Bxx00
J2z (104a)
2Bzz,F00 JzJ,P (104b)+B00 J
2,P + V
e3
1
2(1 cos3) , (104c)
where Bzz,F00 is the rotational constant of the frame and B00 is the effective rotational
constant defined as
Bzz,F0,0 =Bzz0,0
1 (105)
B
0,0 =
Bzz0,0
(1 ) . (106)
The energy expression (104a)(104c) looks similar to that of a rigid symmetric top molecule:
the first term (104a) presents the energy of the overall rotation of the frame and the top and
the second term presents the energy associated with the z component of the total angu-lar momentum of the frame. However, there are additional terms due to internal rotation.
The cross term (104b) arises from coupling between the overall rotation and the torsion.
The first term in (104c) presents the kinetic energy of the top including both and external
rotation while the second term arises from the torsional potential.
3.5.2. Internal Axis Method (IAM): =
In an IAM (Internal Axis Method) axis system [11], the frame and the top rotate at angular
velocities and (1 ) , respectively, about the IAM z-axis. No external angularmomentum is generated since Ie,Tzz + Ie,Fzz (1 ) = 0. The nodal Euler anglesF and T of the two coaxial rotors and that of the entire molecule I are
F = I (107)
T = I + (1 ) (108)I = (1 ) F + T, (109)
where subscript I indicates the IAM. For the sake of simplicity, the components of the
angular momenta are presented in terms of the PAM operators (100)(103). We obtain
Jx,I = (cos ) Jx,P + (sin ) Jy,P (110)
Jy,I = ( sin ) Jx,P + (cos ) Jy,P (111)Jz,I = I
ezzz,I = I
ezzz + I
ezz = Jz,P (112)
J,I = Ie,I = (1
) Iezz = J,P
Jz. (113)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
45/99
45
Again, Jx,I, Jy,I, and Jz,I are the components of the total angular momenta whose quan-tum mechanical operators satisfy the commutation relations (58). Furthermore, Jx,I, Jy,I,and Jz,I all commute with J,I. However, J,I given by Eq. (113) does not represent the
pure torsional angular momentum, but possesses additional contribution from the overallrotation. However, when used in the energy expression, J2,I produces the torsional kineticenergy, and J,I is called torsional angular momentum for the ease of discussion.
In the IAM, the cross term in the inertial tensor Iz in Eq. (94) vanishes, and theHamiltonian (96a)(96c) simplifies to
HI02 = Bxx0,0J
2 +
Bzz0,0 Bxx0,0J2z (114a)
+B0,0J2,I + V
e3
1
2(1 cos3) . (114b)
The term (114a) presents the pure rotation of a symmetric top while term (114b) gives the
energy of an isolated internal rotor in a hindering potential.
3.5.3. Remarks on the Coordinate Conventions. The Hybrid Method
The concepts of the PAM and IAM were originally associated with coordinate systems
of asymmetric rotors, whose rotational-torsional problem is more complex than that of
symmetric tops. The choice of the PAM axis system corresponds to fixing the (x,y,z)
system to the principal axes of the molecule, thus leading to the elimination of the xy,xz, and yz terms of the inertial tensor. However, cross terms with ( = x,y,z) are
preserved, and after inverting the inertial tensor to the tensor, every possible cross termwith JJ (, = x,y,z, ) can appear in the energy expression. The Coriolis-couplingcross terms with JJ ( = x,y,z) can be eliminated by an appropriate rotation of the(x,y,z) axis system, which results in so called Rho Axis Method(RAM) system [23]. Theremaining terms with JJz can then be eliminated by transforming the RAM to PAM.
In the symmetric top case, terms with xy, xz, and yz of the inertial tensor vanish bothin the PAM and IAM, and the torsional axis coincides with the z-axes in both of the axissystems. Therefore, terms PAM and IAM might appear misleading since both the PAM
and IAM are in fact based on the principal axes presentations of the molecule. However, inthe IAM, the cross term Iez vanishes and the associated energy term does not occur in theangular momentum expression of the rotational-torsional energy.
The primary advantage of the PAM is simplicity: the components of the total angu-
lar momentum Jx,P, Jy,P, and Jz,P are defined in a straighforward manner, and thus the
symmetry treatment of the molecular variables is simple. However, in a practical spectrum
analysis, the molecular parameters occurring in Eq. (104a)(104c) are treated as indepen-
dent fitting parameters, and the large cross term (104b) between the torsion and overall
rotation is usually highly correlated with the pure torsional and rotational parameters. Fur-
thermore, at the high barrier limit, the rotational-torsional Hamiltonian does not reduce to
that of a symmetric semi-rigid rotor.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
46/99
46
The advantage of the IAM is the complete separation of the zeroth order rotational and
torsional Hamiltonian, and the rapid convergence of the effective Hamiltonian. However,
the explicit expressions of the angular momenta are complex and a complicated theory is
needed to carry out the symmetry consideration of the molecular variables.The physical difference between the PAM and IAM is the way how the conservation
of the total angular momentum is being treated. In PAM, the torsion generates external
angular momentum whose effect on the rotational energy is then compensated by the cross
term (104b). In the IAM, no torsional angular momentum is generated due to suitable
choice of the coordinate system.
The simplicity of the IAM form of the Hamiltonian (114a)(114b) and on the other
hand, the straightforward PAM presentation (98)(103) of the molecular variables can be
combined in a Hybrid Method (HB) presented in Ref. [14], where the effective torsional
angular momentum is defined as
J = J Jz. (115)
In the HM, the angular momenta are defined as
J = i
P(116)
Jz = i
P. (117)
In the HM, the angular momentum components Jx, Jy and Jz, retain their usual commu-tation relations. However, the effective torsional angular momentum J (115) does not
commute with Jx and Jy as is the case both in the PAM and the IAM. Therefore, the
Hamiltonian terms involving J simultaneously with Jx and/orJy have to be symmetrized
i.e., their anticommutator forms have to be considered.
3.5.4. Schrdinger Equation of a Hindered Internal Rotor
In the following discussion, the (x,y,z) molecule-fi
xed coordinate system is not specifi
ed,and the torsional angle and the nodal torsional angle m are expressed as
= T F (118)m = (1 ) F + T. (119)
Since the coordinate system is unknown, the product of the rotational and torsional eigen-
functions has to be considered. That is
0tr (, , m, ) =
0r
0t = SJk (, ) e
ikmM() . (120)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
47/99
47
The Schrdinger equation of an isolated internal rotor in a threefold hindering potential can
be written as
B00 22 + Ve3 12 (1 cos3)M() = EtM() . (121)With substitutions [9]
2x = 3+ (122)
b =4Et
9B00(123)
s =4V3
9B00(124)
0t [(3+ ) /2] = M(x) , (125)
eq. (121) can be written in the form of the Mathieu equation [9,11]d
dx2M(x) +
b 1
2s
1
2s cos2x
M(x) = 0. (126)
The periodicity of the variables in the Mathieu equation (126) is not yet known since we
assume that the internal coordinate system is not yet selected. A non-periodic solution for
the Mathieu equation can be written as
M() = eifP() , (127)
where P() is periodic in 2, and the parameterf is a real constant chosen so that M()has the periodicity according to the physical situation. Equation (127) shows that the inter-
nal and overall rotation are separable in as much as the rotational-torsional wave-function
can be expressed as a product of a function of the three Eulerian angles and a function ofalone.
Obviously, the wave-function 0tr (, , m, ) must be periodic in 2 with respect tom, which requires that
k [(1 ) n1 + n2 + f(n2 n1)] = n, (128)
where n1, n2, and n are integers. Eq. (128) can be satisfied by taking f = k, thusresulting in wave-function
0tr (, , m, ) = SJk (, ) e
ikmeikP() . (129)
Function P() has periodicity of2 and can be determined by substituting eikP()into Eq. (121). It can be shown [9,11] that the torsional functions Mkv () can be ex-
pressed in terms of the free rotor functions
|k, kf, i =12
eikei(3kf+), (130)
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
48/99
48
where kf is the free rotor quantum number, is the quantum number characterizing thetorsional tunnelling states assuming values = 1, 0, 1.
Let us fix the (x,y,z) coordinate system to the molecule. In the PAM, = 0 and
consequently, P = F which is periodic in 2 by defi
nition. Therefore, the solution ofEq. (121) in the PAM case can be returned to solving the Mathieu equation (126) which
yields free rotor functions
|kf, i =12
ei(3kf+) (131)
as solutions. In the IAM case, = and we obtain
|k, kf, i =12
ei(3kf+k). (132)
The basis set (130) satisfies the orthonormality relation
1
2
Z20
eik
0eiv(3k0
f+0)
eikei(3kf+)
d = k0,kk0f,kf0, (133)
for any . The matrix elements are
-k0, k0f,
0Jn,m |k, kf, i = (3kf + k)n k0,kk0f,kf0, (134)
-k0, k0f,
0
1
2(1 cos3n) |k, kf, i = 1
2
k0
f,kf
1
2k0
f,kf+n
1
2k0
f,kfn
k0,k0,, (135)
where is the Kronecker delta and n = 1, 2, ..In practice, the Schrdinger equation is solved numerically both in the PAM and the
IAM by setting up a Hamiltonian matrix in the free rotor basis. The diagonalization pro-
duces the torsional eigenfunctions
Mk,vT, () =X
kf=AvT,k3kf+ |k, kf, i , (136)
where A
vT,k
3kf+ are the eigenvector components of the torsional wave-functions. In thefi
nalstate (136), the label vT has been adopted to represent the principal torsional quantum num-ber. The quantum numbervT corresponds to a small amplitude labelling of the torsionalstates and assumes values vT = 0, 1, 2, .. in the ascending order of energies.
Neither the effective rotational constant B00 nor the first term V3 of the torsional barrierexpansion can as such provide an easy classification tool to classify molecules according to
their torsional characteristics. However, the Mathieu equation (126) suggests that parame-
ters given in Eq. (124) can do so. Therefore, parameters is also referred to as an effectivebarrier height.
Figures 3, 4, and 5 illustrate torsional energy level structure and probability densities
with s = 7, s = 38, and s = 94, respectively. Value s = 7 illustrates an imaginary case,
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
49/99
49
while values s = 38 and s = 94 correspond to CH3SiD3 (Papers I and IV) and CH3CF3(Paper II), respectively.
In Figures 3, 4 and 5, the torsional potentials are shown with dark solid sinusoidal
curves. Energy levels labelled with vT = 0,.., 6 are illustrated with thick dashed/dottedhorizontal lines, whereas the probability density curves are plotted with thin dashed/dottedlines. The probability density is obtained by taking a square of the torsional wave-funtion
at each torsional angle and presents the probability offinding the torsional oscillator at a
specific torsional angle.
Case s = 7 illustrated in Figure 3 represents a low barrier case. The kinetic term in thetorsional Hamiltonian (121) dominates, and the energy level structure is similar to that of
a free rotor. Furthermore, the splittings are large due to high torsional tunnelling rate.The free rotor expansion (136) representing the torsional wave-function converges rapidly:
in this case 7 terms are required to obtain the accuracy of 106 cm1 in eigenenergy.The energy corresponding to the lowest state with vT = 0 is well below the top of the
barrier, and the probability density is similar to that of the harmonic oscillator. At vT = 1,the probability density shows two maxima characteristic to the harmonic oscillator but the
curve shape is flattened, and the splittings are of the order of10 cm1 indicating thetorsional tunnelling. At vT > 1, the quadratic separation of the energy levels typical to freerotor dominates, and the probability density is not well localized in torsional angle. Now,
the splittings are more than 100 cm1.Case s = 38 illustrated in Fig. 4 falls to an intermediate barrier category. The lowest
torsional states vT = 0, 1, 2 are almost equally separated, and the probability distributionplot is very similar to that of a harmonic oscillator. Now, 15 terms in the wave-functionexpansion (136) are needed. The splittings forvT = 0, 1, 2 are 0.1 cm1, 0.7cm
1
and 10 cm1
, respectively. Forv > 2 6, the splittings are more than 40cm1, and the probability offinding the oscillator inside the barrier i.e., at angles 60, 180,and 240, is now substantially larger.
Case s = 94 exemplified in Fig. 5 shows a high barrier case. The splittings for levelsvT = 0 4 are 0.0001 cm1, 0.001 cm1, 0.01 cm1, 0.5 cm1, and 5 cm1,respectively. The levels with successive values ofvT are virtually evenly spaced indicatingthe harmonic oscillator nature, and the probability distributions are similar to those of the
harmonic oscillator. In this case, 21 terms are required in the wave-function expansion(136) to obtain the sufficient accuracy. ForvT = 5 and vt = 6, the splittings increaserapidly to 50 cm1 and the free rotor nature of the probability density takes over.
The classification of the torsional characteristics into the three cases according to the
effective barrier height s is given only for discussion purposes and does not follow anyknown classification given in literature. The main purpose of the classification is to il-
lustrate the cases between a virtual free internal rotor and a virtual harmonic oscillator.
It should be pointed out that the free rotor functions (130) provide an efficient basis set
for characterizing motion in a periodic potential from the free rotor limit to the harmonic
oscillator limit.
The energy level systems exemplified in Figs. 3, 4 and 5 represent the simplest case
where no small amplitude vibrational quanta are excited. However, when vibrational
states are excited, each vibrational excited state holds a stack of torsional levels with
vT = 0, 1, 2,.., whose details primarily depend on the effective rotational constant B
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
50/99
50
50 100 150 50 100 15050 100 150
0
200
400
600
800
1000
Energy(cm
-1)/Tor
sionalpropabilitydensity
66
6
55
5
44
4
33
3
22
2
111
000
K=1,=+1K=1,=0K=1,=-1
Torsional an le
Fig. 3. Torsional energy level diagram and the probability density plots in the low barrier case
for K = 1, = 1, 0, +1. The torsional energies are indicated with thick lines, which also
define the baseline for the probability density curve. The vT quantum number is given for each
energy level.
and the torsional potential V3 in the specific vibrational state. As a result, a complex andperturbed energy level system is generated.
3.6. Molecular Symmetry
Symmetry properties of molecules together with the mathematical machinery of the group
theory make up an efficient tool in studying molecular systems. In molecular spectroscopy,
group theory can be used to simplify considerably the theory of the quantum states and the
expressions of the Hamiltonian and transition moments.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
51/99
51
50 100 150 50 100 15050 100 1500
200
400
600
800
1000
1200
Energy(cm
-1)/Tor
sionalpropabilitydensity 6
6
6
55
5
44
4
33
3
22 2
111
000
K=1,=+1K=1,=0K=1,=-1
Torsional an le
Fig. 4. Torsional energy level diagram and the probability density plots in the intermediate
barrier case for K = 1, = 1, 0, +1. For other details, see Fig. 3.
Fundamentally, the symmetry properties of functions k (), m () , and P () dis-criminate whether a matrix element associated with functionsk (), m () , and P ()given by integral
Z
k ()P ()m () d (137)
vanishes. In the case of a vibrating and rotating molecule, functions k () representthe vibration-torsion-rotation wave-functions that depend on a set of coordinates and a
set of quantum numbers k and m. Function P () is a general vibration-torsion-rotationoperator. In group theory language, integral (137) may have non-vanishing values, when
the direct product of representations spanned by k (), m () and P () contains atotally symmetric representation. Furthermore, the Hamiltonian has to be invariant to time
reversal and Hermitian conjugation.
8/3/2019 Jyrki Schroderus- Internal Rotation in Symmetric Top Molecules
52/99
52
50 100 150 50 100 15050 100 1500
200
400
600
800
1000
1200
Energy(cm
-1)/Tors
ionalpropabilitydensity
66
6
55
5
444
333
22 2
111
000
K=1,=+1K=1,=0K=1,=-1
Torsional an le
Fig. 5. Torsional energy level diagram and the probability density plots in the high barrier case
for K = 1, = 1, 0, +1. For other details, see Fig. 3.
A symmetry group consists of a set of operations to which the energy of the system
is invariant: these operations are called elements. The elements of a Complete Nuclear
Permutation Inversion (CNPI) group
Top Related